Integrand size = 98, antiderivative size = 27 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=e^x \left (-x+\frac {6 x}{-2+e^{\frac {1}{3} x (2+2 x)}}\right ) \]
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\[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=\int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-16 (1+x)-e^{\frac {4}{3} x (1+x)} (1+x)-2 e^{\frac {2}{3} x (1+x)} \left (-5-3 x+4 x^2\right )\right )}{\left (2-e^{\frac {2}{3} x (1+x)}\right )^2} \, dx \\ & = \int \left (-e^x-e^x x-\frac {8 e^x x (1+2 x)}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2}-\frac {2 e^x \left (-3-x+4 x^2\right )}{-2+e^{\frac {2}{3} x (1+x)}}\right ) \, dx \\ & = -\left (2 \int \frac {e^x \left (-3-x+4 x^2\right )}{-2+e^{\frac {2}{3} x (1+x)}} \, dx\right )-8 \int \frac {e^x x (1+2 x)}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2} \, dx-\int e^x \, dx-\int e^x x \, dx \\ & = -e^x-e^x x-2 \int \left (-\frac {3 e^x}{-2+e^{\frac {2}{3} x (1+x)}}-\frac {e^x x}{-2+e^{\frac {2}{3} x (1+x)}}+\frac {4 e^x x^2}{-2+e^{\frac {2}{3} x (1+x)}}\right ) \, dx-8 \int \left (\frac {e^x x}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2}+\frac {2 e^x x^2}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2}\right ) \, dx+\int e^x \, dx \\ & = -e^x x+2 \int \frac {e^x x}{-2+e^{\frac {2}{3} x (1+x)}} \, dx+6 \int \frac {e^x}{-2+e^{\frac {2}{3} x (1+x)}} \, dx-8 \int \frac {e^x x}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2} \, dx-8 \int \frac {e^x x^2}{-2+e^{\frac {2}{3} x (1+x)}} \, dx-16 \int \frac {e^x x^2}{\left (-2+e^{\frac {2}{3} x (1+x)}\right )^2} \, dx \\ \end{align*}
Time = 3.82 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=-e^x \left (1-\frac {6}{-2+e^{\frac {2}{3} x (1+x)}}\right ) x \]
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Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
risch | \(-{\mathrm e}^{x} x +\frac {6 x \,{\mathrm e}^{x}}{{\mathrm e}^{\frac {2 \left (1+x \right ) x}{3}}-2}\) | \(23\) |
norman | \(\frac {8 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x \,{\mathrm e}^{\frac {2}{3} x^{2}+\frac {2}{3} x}}{{\mathrm e}^{\frac {2}{3} x^{2}+\frac {2}{3} x}-2}\) | \(37\) |
parallelrisch | \(-\frac {{\mathrm e}^{x} x \,{\mathrm e}^{\frac {2}{3} x^{2}+\frac {2}{3} x}-8 \,{\mathrm e}^{x} x}{{\mathrm e}^{\frac {2}{3} x^{2}+\frac {2}{3} x}-2}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=-\frac {x e^{\left (\frac {8}{3} \, x^{2} + \frac {17}{3} \, x\right )} - 8 \, x e^{\left (2 \, x^{2} + 5 \, x\right )}}{e^{\left (\frac {8}{3} \, x^{2} + \frac {14}{3} \, x\right )} - 2 \, e^{\left (2 \, x^{2} + 4 \, x\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=- x e^{x} + \frac {6 x e^{x}}{e^{\frac {2 x^{2}}{3} + \frac {2 x}{3}} - 2} \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=-\frac {x e^{\left (\frac {2}{3} \, x^{2} + \frac {5}{3} \, x\right )} - 8 \, x e^{x}}{e^{\left (\frac {2}{3} \, x^{2} + \frac {2}{3} \, x\right )} - 2} \]
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Timed out. \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=\text {Timed out} \]
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Time = 0.44 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^x (-16-16 x)+e^{x+\frac {2}{3} \left (2 x+2 x^2\right )} (-1-x)+e^{x+\frac {1}{3} \left (2 x+2 x^2\right )} \left (10+6 x-8 x^2\right )}{4-4 e^{\frac {1}{3} \left (2 x+2 x^2\right )}+e^{\frac {2}{3} \left (2 x+2 x^2\right )}} \, dx=\frac {6\,x\,{\mathrm {e}}^x}{{\mathrm {e}}^{\frac {2\,x^2}{3}+\frac {2\,x}{3}}-2}-x\,{\mathrm {e}}^x \]
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